Broken Symmetry

Example of How Evidence can Undermine Bad Reasoning

Provided by Karl Rove in today's WSJ:

Mr. Obama's changes could overload Chief of Staff Rahm Emanuel. As power that was once diffused to cabinet officers is centralized in the White House, Mr. Emanuel will have to make more decisions and referee more turf wars than his predecessors. This will test his skills and likely inject chaos into the policy process as he prioritizes the decisions he can reasonably make.

Mr. Obama's tendency to work late into the night will also pose problems. Politico.com reports that the White House staff is "preparing for a return to long nights, heavy weekend shifts." Requiring a senior staff that meets at 7:30 a.m. to work until 11 p.m. or 12 a.m. will quickly cause burnout and diminish the quality of advice and oversight.

How is spending more time and listening to more points of view going to centralize decisionmaking and diminish the quality of advice and oversight? Rove gives us more actual insight into the management problems of the Bush administration through this criticism of Obama than we've gotten from any of Bush's critics.

How Rochester Solved a Prisoner's Dilemma

The Tech Transfer blog reports:

The program is the brainchild of university president Bill Destler, who implemented it shortly after taking office in 2007. Destler argued that corporations are often reluctant to enter into certain R&D agreements with universities because of the anticipated wrangling over IP rights. Under the new program, RIT relinquished all rights to future intellectual property generated in exchange for one-time, modest up-front payments. The work is carried out over the course of a year at a fixed price — typically under $100,000 — which the company pays in quarterly installments. The school also provides partnering companies with a grad student and a faculty adviser to work on specific research projects. RIT retains the right to publish scientific results after the sponsor has had a pre-negotiated amount of time to review the discoveries, and also gets the right to continue using the IP in future research.

The first research university to adopt this model for R&D with potential clinical relevance is going to see its endowment expand faster than it contracted in 2008.

This could get interesting

From the President's blog:

I know that some are skeptical about the size and scale of this recovery plan. I understand that skepticism, which is why this recovery plan must and will include unprecedented measures that will allow the American people to hold my Administration accountable for these results. We won’t just throw money at our problems - we’ll invest in what works. Instead of politicians doling out money behind a veil of secrecy, decisions about where we invest will be made public, and informed by independent experts whenever possible. We’ll launch an unprecedented effort to root out waste, inefficiency, and unnecessary spending in our government, and every American will be able to see how and where we spend taxpayer dollars by going to a new website called recovery.gov.

Thank you to Michael A. Gollin

It's been a pleasure to have Michael A. Gollin as a guest here at Broken Symmetry, and I want thank him for sharing his thoughts with readers. Please do make sure to grab a copy of Driving Innovation so that you can benefit more from his insights.

I'm glad that Michael brought up Lincoln again, because I think we agree that his insights on the patent system, although nearly 150 years old to the day, remain a sound basis for our new President to rely upon in considering various patent reforms. In the lecture he gave on Inventions and Discoveries in Jacksonville, Illinois on February 11, 1859, Lincoln declared that:

All nature -- the whole world, material, moral, and intellectual -- is a mine; and, in Adam's day, it was a wholly unexplored mine. Now it is the destined work of Adam's race to develope, by discoveries, inventions, and improvements, the hidden treasures of this mine.

In Driving Innovation, in addition to the metaphor of an engine, you chose to describe the flow of innovation as an "innovation forest." The innovation forest is naturally closer to the ecosystem I had in mind in my reply to your first post. Innovation is not a sterile, individualistic enterprise. Ideas evolve healthiest and fastest in an organic, collaborative and competitive scramble in which new ideas are given resources and protection and older ideas are left alone to be used as a resource directly, or to provide indirectly through their decay into the commons a rich soil for new growth.

In fact, I believe your metaphor is an improvement on Lincoln's mine. From our perspective 150 years on, it is obvious that the natural world (including minerals), are not the bottomless pit that they might have seemed to us in the mid-19th Century. Indeed, there is now a broad consensus that our use of natural resources must be made with an eye toward their soon exhaustion.

How fortunate we are the ideas are more like trees than like minerals! When our minerals and fossil fuels are exhausted, there will be no way for us to replace them in our generation. But with an eye toward the future, and with careful planning, we could someday ensure that we do not use up our living resources before they can be replaced.

Beginning around the time of Lincoln, the incentives of the patent system helped to grow a redwood forest of ideas in the United States. On my view, we have been clearcutting from that forest for the last few years. There is nothing wrong with harvesting part of the bounty of what nature has provided us. But such harvesting should be done with an eye toward leaving enough for the forest to flourish on into the future, so that others may enjoy its benefits after us.

I like the idea that as a patent lawyer, I am a steward of that kind of national and natural treasure. I like your idea of a National Innovation Council, which could assist with determining what parts of the forest are in need of more fertilizer, and which need a brush fire to clear out the dead waste. Perhaps the innovation forest has grown too complex to be regulated under a single set of rules. We would not expect a redwood forest to grow well under the same conditions as a tropical jungle. So why should we expect innovation forests to be any different? I worry, however, that the best forestry experts will not be around to serve on such a council. Only time will tell.

Thanks again for sharing your thoughts.

Guest Post: Reply by Michael A. Gollin

On this inauguration day of another Illinois President, it is worth reflecting on patentee Abe Lincoln's wise perspective on innovation and the ideal role of patents. In this light, I view your concerns about broken symmetry as a measure of the gap between the ideal and the real, between theory and practice.

You suggest that the cost of prosecuting patents has decreased, but my clients would disagree, noting that over the past couple of years we face a heavier burden to satisfy heightened standards for patentability imposed by the courts and the PTO. Litigation costs remain very high, not so much because of a lack of qualified lawyers, but because of uncertainty about how to interpret patent claims, so that many complex issues of law and fact remain unresolved after extensive discovery. Your third example, Bayh-Dole Act patenting by universities, does not in and of itself raise transaction costs of clearing IP ownership -- rather, I might argue that many of the licensing specialists in technology transfer offices actually streamline licensing because of their expertise and know-how.

Your concept of a "rich ecosystem" of innovation is quite sound. And yes, in Driving Innovation, I describe how IP's oscillation between access and exclusivity may operate as a dynamo or motor for the innovation cycle. (In my book, I occasionally mix the metaphor of a biological ecosystem with that of a machine.) In the innovation cycle, every creative person benefits from access to a wealth of information from which new ideas arise. The creator can then control dissemination of the new ideas via IP rights, exerting control over their use and duplication. This is the mechanism by which companies pool the innovations of employers, and individuals can attract the investment needed to put the idea to work as a product or service or published work. Exclusivity and control lead to investment and dissemination, and thus, heightened access by the public, as the product or service or information becomes widely available. Finally, the fruits of the new idea become widely accessible by society, and can be used to feed the next round of innovation. Access feeds exclusivty, and exclusivity feeds access, like the alternating current in an electric motor. In this view, the key is to get the balance right at each phase of the cycle, and as you say, between big and small players. Because ultimately, society at large benefits when creators and the public have both sufficient access and sufficient exclusivity to make innovation worthwhile.

Coming back from theory to your practical comments, it would certainly be nice to have more flexibility in presenting inventions involving images, whether color "heat maps" of biological data or image processing software. However, a reasonable consistency in the presentation of material is a compromise most of us can accept. Also, you may have hit on one of the Dudas' administration's suceesses, in which the public access PAIR system and electronic filing have finally begun to work well. And (you've got quite a few practical ideas, Michael) it would certainly improve markets in patents if more information about licensing terms was publicly available. But this goes again to the delicate balance between the innovator's right to restrict access and the public's benefit to having access. There are many good grounds for a patentee to keep the economic terms of an agreement confidential, thus blending patent protection for technology inventions with trade secret protections for business information. I didn't find the details of Lemley and Myhrvold's paper -- perhaps this is another theory that does not yet have a viable mechanism for putting it into practice.

Most of the day-to-day projects that arise in my representation of clients involve some aspect of these questions -- how to access a computer program or biological reagent or use a piece of hardware or a song; or how to protect a new medical device or branded product from copyists. Although a brief exchange via blog is inadequate to the task, all of intellectual property -- theory and practice -- can be understood as part of this larger balance between exclusivity and access.

How can any reasonably curious person be content with this explanation?

From the Fama/French forum:

Every market determined price is an equilibrium price that should take account of all information available at the time the price is set. (This is the definition of market efficiency.) But things inevitably change, and equilibrium prices change along with them. All we can say about the recent market turmoil is that the volatility of information and its implications for forecasts of profitability must be quite high.

Is anybody else flabbergasted that leading economists could content themselves with such a poor theory of market cycles? Or have all economists really convinced themselves that business cycles are just part of the demon haunted world of non-economists?

IP as Solution to Market Failure: a Few Responses to Mr. Gollin

What an inspiration to hear that Senator Bayh is still so active in his 80s!  His first-hand knowledge of and experience with the last major round of patent reform is even more important now, and it's great to hear that he's still keeping a hand in the game.

Your first post raises many very interesting points, and I want to try and draw you out a little more on some of them.

First, you are rather too charitable in attributing to me the observation of a need for what you call a shared model of innovation.  But I very much like the vision you have of IP striking a balance between self-interest in exclusivity and self-interest in access.  Indeed such tensions have always existed between innovators and the beneficiaries of their work, and I think we both agree with Lincoln that the IP system has over time proved to be an ingenious way of striking that balance.

I think it's intriguing to consider, however, how much the identities of the parties to these types of negotiations have changed over the past few decades.  You give "two individuals, two companies, or two nations" as examples of the parties to such a negotiation, examples that tend to suggest (and perhaps this was not your intent) relative equality of position -- i.e., relatively symmetrical costs and benefits on each side.  In this sense, important symmetries have been broken within IP law over the past few decades.  These broken symmetries (I can't help using the metaphor, which fits so well here) are largely an unanticpated consequence of many improvements that have been made to the IP system (and I'll stick to patents here, although I see similar trends in copyright):

• Costs of prosecution have generally been stable or decreasing, putting patent rights within grasp of individuals and foreign entities
• Costs of litigation have skyrocketed as the number of litigations and litigation defendants has increased much faster than the supply of lawyers qualified to handle the work
• Bayh-Dole, at the same time that it opened up new sources of private funding for universities, has also raised the transactions costs of clearing IP ownership

As a result, the patent system is no longer adequately described as a competition among large corporations, which occasionally erupts into patent wars, and is settled by cross-licensing treaty.  Rather, an incredibly rich ecosystem is emerging in which individual inventors, R&D labs (public like Rambus, private like Intellectual Ventures, or private like universities), private equity funds (RPX, for example), public investment funds (like Acacia), and the public and private corporations that want access to IP owned by the preceding entities and are net payers under the patent system.  The richness of the system itself is ultimately what merits a more sophisticated theory, such as that presented in your book.

With that lead in, I wonder if you couldn't say a little about your concept of "IP as a motor for the innovation cycle" and how that vision of IP is more useful in the rich ecosystem that has developed?  If the problem is no longer one of simply reallocating rights among large corporations, and smaller entities can't effectively negotiate with large corporations without strong IP rights, then aren't we effectively threatening innovation by limiting damages and rights to injunctions?

I will leave that one big question for you, but I wanted also to address briefly your other comments on the three reform proposals I raised:

The idea of making patent disclosures more user friendly to people other than patent attorneys (and leveraging new technology to do so) is one that occurred to me after seeing that an inventor has started a company called Planet Eureka, the business of which is to provide a website for inventors to post descriptions of their inventions in an attempt to attract licensees.  Planet Eureka runs a workshop, which coaches the inventors on how to pitch their work using multimedia, and then has a catalog of the various inventions up on their website for potential licensees.  I hate to say it, because Planet Eureka does excellent work, but the PTO should ultimately put this kind of website out of business!  The patent system isn't working quite right if inventors are willing to go that route rather than dislcose to the public through the PTO in exchange for exclusive rights.  Ideally, the PTO would collaborate with Planet Eureka to make its services available to all inventors and potential buyers.

Second, Jon Dudas deserves kudos for overseeing the complete overhaul of the PAIR and EFS system.  In this regard, the progress made at the PTO during his tenure substantial and excellent.  In fact, the PAIR system makes determining patent ownership (except, perhaps, for some pesky corporate and bankruptcy law issues) cheap and fast.  In that sense, an ownership clearinghouse of the kind that IDEA appears to have in mind already a reality.  I'm probably missing some nuances of the proposal.

But what I had in mind was the proposal by Mark Lemley and Nathan Myhrvolds that all buyers and sellers of patents should be required to disclose the terms of all licenses and assignments of patents to the government, much as public corporations are required to report their financial statements to the SEC.  Most people flinch at this idea when they hear it because that information absolutely must be kept confidential in the world we live in now.  The question is whether we might all be better off if nobody was allowed to keep it private.  A long time ago I wrote about how such a rule is like the NHL's decision to require players to wear helmets long ago.

I'll get back to antitrust if I see the opportunity later, but suffice it to say that the FTC handling of the Rambus and Schering-Plough cases makes it obvious that some of the folks there don't really understand how IP can promote competition.

Guest Post: Michael A. Gollin

Thank you for your kind invitation. It is a pleasure to read your insightful comments and I hope to see what your readers think.

Let me begin by pointing out that Sen. Bayh is still in office – our offices at Venable – several days a week. Not bad for a man in his 80s. I hope that I am able to do the same because I am sure that in 30 years the practice of IP law will be even more fascinating than it is today.

I agree completely with your first point, that there is a need for a shared model of innovation that goes beyond self-interest. (Speaking of self-interest, I must also agree reflexively with any suggestion that Driving Innovation should be required reading.) In essence, my view is that the role of IP systems is to balance the selfish interest in exclusivity and access. That is, I want to limit your access to my innovation while having full access to yours, but you want to have full access to my innovation, while limiting my access to yours. This gives rise to an inevitable tension, of course. This tension exists whether you and I refer to two individuals, two companies, or two nations. In any given situation involving innovation, whether a negotiation, a lawsuit, lobbying, or international relations, there will be one party favoring more protection, and one favoring less. IP laws determine where the balance should be struck – not for the benefit of those two parties only, but for society at large.

I am not an economist, and Driving Innovation focuses mostly on practical steps you can take to pursue your organizational (or national) goals. However, my attitudes do rest on some fundamental economic assumptions, including those of Adam Smith, and Joseph Schumpeter's work on creative destruction as a model for how innovation moves through and reshapes society. Innovation means that there will always be winners and losers, haves and have-nots. That creates political tensions, too.

Now is a great time to think creatively about patent reform, and reform of copyright and trademark and other IP rights. These laws have changed radically over the centuries, and no doubt will evolve into the distant future as well.

I haven't given much thought to your first suggestion, a revision of patent disclosure rules, but it would make sense to allow more flexibility in how inventions are exemplified in patent applications. After all, we long ago eliminated the requirement for inventors to provide a miniature working model of their inventions. (A few of those intricate old models are exhibited in the Smithsonian American Art Museum, in the former Patent Office, across the street from where I sit.) One constraint in opening up patent rules might be the need to present information in a manner that can be published, translated, and understood internationally. But any reform that can reduce the time and cost required to prepare a patent application would be welcome.

There is broad support for a clearinghouse of ownership information, and IDEA magazine a few years back reported on a symposium of a national (or international) registry that could be used to support securities in IP – like a registry of deeds. That is the easy part. Placing a value on IP rights is much more complex, as it requires an understanding of the scope of the rights, their duration, their geographical extent, their relevance to a particular market, and the size of that market. So the ownership clearinghouse is a much simpler objective.

I am not so troubled by antitrust requirements, in most cases. The ascendancy of the rule of reason as to patents and antitrust laws is generally a good thing, and economists seem to recognize that patents can promote (rather than hinder) competition. Here, too, there is no simple answer. It will never be easy to balance the tension between exclusivity (and short term limits on competition) against increased access over the long term, as more innovations become available. Yes, there are more patents than ever before, and more works restricted by copyright. But there are also many more inventions and creative works more readily available than ever before.

My main proposal is quite simple -- that people should master the basics of intellectual property and its dynamics and put that knowledge to work to promote the interests of their organizations and their nations. Identify your goals, assess your internal resources, learn your competitive environment, and come up with a simple, long-range plan to protect your innovations and to gain access to the innovations you require. Practical solutions exist to many problems that people try instead to solve with policy reform.

At the other extreme, if you want pie in the sky, try this: the US should establish an independent National Innovation Council that would determine, for each type of IP right, whether the laws are too strong (with too much exclusivity), too weak (with too little exclusivity), or just right. An analogy is the Federal Reserve's responsibility in determining whether interest rates are too high or too low. The findings of the NIC could be used to help Congressional staffers find better solutions than those that come out of partisan duels. The NIC could also provide objective information about innovation productivity and the implications of various reforms and how they would impact different industries. This information could also help shape national innovation policy, investments in research, and international trade policy.

I'm glad you the book triggered these productive ideas. But that's all for now.

Introduction for Guest Blogger Michael A. Gollin

As a fun way to kick off 2009 for Broken Symmetry readers, I am welcoming author, patent lawyer, and professor Michael A. Gollin to guest blog on Broken Symmetry. Michael has had a distinguished career, which includes almost ten years of teaching Strategic Management of Intellectual Property at Georgetown.  Recently, he authored Driving Innovation: Intellectual Property Strategies for a Dynamic World, which was released by Cambridge University Press in February 2008.  To give readers a sense of the potential importance of this book to policymakers, the introduction was provided by former Senator Birch Bayh of Bayh-Dole Act fame.  (If only Senator Bayh were still in office!)

Beyond the pleasure of his company, I wanted to invite Michael to guest blog because 2009 marks a pivotal moment in the history of intellectual property.  Patent reform efforts that can only be called misguided were narrowly defeated in 2008.  Many in Congress and in the Obama administration are aware that there are problems with the current patent system, but the proposals for change are all over the map.  If anything, the guiding principle behind most proposals seems to be simple self-interest.  What is needed badly at this moment is a comprehensive economic theory of innovation, and of how the patent system can work to promote innovation.  Mr. Gollin's book should be required reading for anybody involved with patent reform.

The most important reason why Mr. Gollin's book succeeds at explaining how and why the patent system can be an effective means for promoting the progress of science and engineering is dynamics.  Policymakers have understood how economic theory is important to both patent and copyright law for a long time.  But many of the most prominent theorists of law and economics have overlooked how static models of equilibrium (all related in one way or another to the Arrow-Debreu model) fail to accomodate change over time.  Whereas such a failure might have been tolerated in older industries and prior to globalization, it cannot in any industry in which innovation drives growth.  More importantly to present purposes, it cannot adequately explain how and why a patent system can benefit both inventors and society at large.

But rather than summarizing the numerous principles of dynamic economic theory that are explained and applied to intellectual property in his book (read the book!), I want to take this opportunity to engage him in a discussion of patent reform.  What subsantive or procedural changes would be most beneficial (or disadvantageous) to IP and antitrust law?  How does a dynamic theory of IP and competition inform our policy analysis in ways that carry us beyond the standard arguments?

Here are a few areas that I can identify:

• Complete overhaul of patent disclosure requirements: The 19th Century specifications and drawings now filed should be supplanted by full multimedia so that inventors can explain their work in their own words to others in their field using full-color video and graphics.  Let the lawyers worry about claiming.
• Clearinghouse for Ownership and Licensing Information: Any member of the public should be able to determine quickly and easily (for example, through an Internet search) the ownership and market price of a patented technology.  The reason there is not a liquid market for IP rights is because valuations are difficult when price information stays private.
• Complete overhaul of antitrust rules as applied in cases involving IP:  Recent FTC actions suggest a grave misunderstanding of the pro-competitive role of patents.  Patents raise barriers to entry in the markets for patented products and processes, but patents also lower barriers to entry in the markets for R&D.  Because these two competitions take place at different times -- often decades apart -- the net pro-competitive effect of patenting is not grasped by some antitrust decisionmakers.

Michael, am I wrong in thinking that the dynamic theory presented in your book might show favor on these kinds of proposals?  Are there others that you would favor more?

Hal Varian on How to Stimulate Demand

Easily the best commentary on how government can effect an economic recovery that I've read in a while:

There are worthwhile public infrastructure projects; the trick is to find them and fund them promptly. One possible plan is to set up an independent commission to prioritize public investment projects, and then subject the plan to a single up-or-down vote in Congress. One further warning about government stimulus: It makes little sense for the federal government to spend more if the states are forced to spend less. A significant part of the increase in federal government spending should be transfers to the states in order to keep basic government services available at the state and local level.

RPX Closes Deal with Acacia

For "light valve" technology.  See press release here.

This must be one of the first deals closed by RPX since its launch late last year.

The deal is not unexpected given what has been publicly disclosed about RPX's business model. It will be interesting to see whether other intermediaries close deals for the same portfolio subsequently.

2009: Year of the OLED?

How about 1 inch thick displays with 2,000,000 to 1 contrast ratio, and less power consumption than LCDs: See the LG OLED at CES 2009.

I've been watching this technology develop for almost ten years. It's going to be a big deal when it rolls out in mass production. A device like the Kindle with an OLED is going to replace the books I carry around with me everywhere.

Global Consciousness

Another fascinating excerpt from the Edge symposium, this one by Danny Hillis:

In 1851, Nathaniel Hawthorne wrote, "Is it a fact — or have I dreamt it — that, by means of electricity, the world of matter has become a great nerve, vibrating thousands of miles in a breathless point of time? Rather, the round globe is a vast head, a brain, instinct with intelligence!" He was writing about the telegraph, but today we make essentially the same observation about the Internet.

One might suppose that, with all its zillions of transistors and billions of human minds, the world brain would be thinking some pretty profound thoughts. There is little evidence that this is so. Today's Internet functions mostly as a giant communications and storage system, accessed by individual humans. Although much of human knowledge is represented in some form within the machine, it is not yet represented in a form that is particularly meaningful to the machine. For the most part, the Internet knows no more about the information it handles than the telephone system knows about the conversations that take place over its lines. Most of those zillions of transistors are either doing something very trivial or nothing at all, and most of those billions of human minds are doing their own thing.

If there is such a thing as a world mind today, then its thoughts are primarily about commerce. It is the "invisible hand" of Adam Smith, deciding the prices, allocating the capital. Its brain is composed not only of the human buyers and sellers, but also of the trading programs on Wall Street and of the economic models of the central banks. The wires "vibrating thousands of miles in a breathless point of time" are not just carrying messages between human minds, they are participating in the decisions of the world mind as a whole. This unconscious system is the world's hindbrain.

The whole of this one should be read for the full effect.

Anybody Want to Launch a Journal of Negative Results?

Nassim Taleb also nails a good point in the Edge symposium. A professor I had as an undergrad used to say that science could really use a journal of negative results:

Unfortunately such lack of rigor pervades the place where we expect it the least: institutional science. Science, particularly its academic version, never liked negative results, let alone the statement and advertizing of its own limits — the reward system is not set up for it. You get respect for doing funambulism or spectator sports –following the right steps to become the "Einstein of Economics" or the "next Darwin" rather than give society something real by debunking myths or by cataloguing where our knowledge stops.

UPDATE: I see that we now have one -- for Evolutionary Biology, natch.

On Why Economists and Physicists Need to Take Time Seriously

Lee Smolin nails this in an Edge symposium:

I used to think that the goal of physics was the discovery of a timeless mathematical equation that was isomorphic to the history of the universe. But if there is no Platonic realm of timeless mathematical object, this is just a fantasy. Science is then only about what we can discover is true in the one real universe we find ourselves in.

More specifically, this view challenges how we think about cosmology. It opens up new ways to approach the deepest questions, such as why the laws we observe are true, and not others, and what determined the initial conditions of the universe. The philosopher Charles Sanders Pierce wrote in 1893 that the only way of accounting for which laws were true would be through a mechanics of evolution, and I believe this remains true today. But the evolution of laws requires time to be real. Furthermore, there is, I believe, evidence on technical grounds that the correct formulations of quantum gravity and cosmology will require the postulate that time is real and fundamental.

But the implications of this view will be far broader. For example, in neoclassical, economic theory, which is anchored in the study of equilibria of markets and games, time is largely abstracted away. The fundamental results on equilibria by Arrow and Debreu assume that there are fixed and specifiable lists of goods, and strategies, and that each consumer’s tastes and preferences are unchanging.

But can this be completely correct, if growth is driven by opportunities that suddenly appear from unpredictable discoveries of new products, new strategies, and new modes of organization? Getting economic theory right has implications for a wide range of policy decisions, and how time is treated is a key issue. An economics that assumes that we cannot predict key innovations must be very different from one that assumes all is knowable at any time.

Bleg: Geometry of Algorithms?

On a walk with a mathematician friend today we fell to talking about the value that geometric insights have had in solving difficult problems. The general thought is that great analysis is useful in solving difficult problems in well-understood systems. But geometric understanding has often been necessary to setting up the analysis properly in the first place. When I think of great physics, for example, an extraordinary ability to visualize geometry seems to run as a common thread through the best theory.

The other topic of conversation was algorithmic theory. We've been putting together algorithms for a long time -- every mathematical proof (whether true, false, or undecidable) -- is an algorithm. Algorithms are more general than equations.

So naturally we were talking about how one might go about setting up a geometry of algorithms. The connection between geometry and analysis (both real and complex) would be a subset of a more general geometry of algorithms -- viz., the subset in which real and complex functions are associated with a geometric manifold.

But let's say I wanted to start visualizing algorithms in two or three dimensions. Is there a theory for this? I remember from years ago that there are many different sorting algorithms, with quicksort being one of the fastest. Is there a systematic way to represent these algorithms in two or three dimensions and bring geometric intuition to bear on their similarities and differences?

The naive method that we talked about would involve taking a turing tape of 0s and 1s, breaking it into finite segments (of, say, 16 bits), and then mapping each segment into a two dimensional discrete grid of 256 points. An algorithm, then, would be a trajectory through that discrete space. This is unlikely to produce geometrical insights, however.

My friend insists that this is what graph theory is all about. I need to learn more graph theory. But if there are any readers that can point me towards a web page, paper, or book that provides an introduction to this type of theory, I'd be much obliged.